3 Answers
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The metagame is established generally through the thousands of games played. Usually, teams will get together and decide to try a team composition and success will lead them to keep trying it. Obviously, over just a few games everyone's performance will vary widely and no meta will save you on your bad days...

...But over thousands of games with hundreds of thousands of players, individual skill becomes averaged out and what you're left with is what strategies tend to work. Now, we need to distinguish a strategy from a meta to make a lot of sense here. I'll do it very briefly: The metagame is a framework in which you insert individual pieces (champions and builds, where builds are item/mastery/rune combos) that fit the framework. In our current state, the "meta" is usually a high-survivability (but not usually tank) in top who can solo and scales well with items and experience; mid is a character that scales best with experience; bot is a character who scales best with items and needs either baby-sitter or someone to setup kills; the jungler is highly variable, but often is prized for their ganking abilities. So, take any "pieces" that fit inside that framework, and you have a particular strategy within the meta. For instance, an AoE heavy composition is a strategy within the current meta.

Strategies are handed down from pro players (usually, but not always), but the meta is determined by all of us. It's less a "choice" that people get to make and more a mathematical optimization realized over a ridiculous number of games (millions). Metas will usually stabilize without some dramatic changes in terms of patches and champions, which is what has happened in LoL. More casual players will tend to emulate what they see the pro players do, which makes sense in terms of the number of games played per person.

The beginnings of a meta can start anywhere. It can be some 1000 elo troll playing something or some top pros spending time theory crafting.

However, for it to become popular and widely use, it either has to become instantly popular like with sunfire Garen or solo lane Malphite a long ways back or it gets used by pros like the AD+Support bot set up that NA borrowed from Europe.

From here, there will always be a constant flux in meta with new patches and champions but it usually revolves around these 4 central ideas.